Axiomatic Syntax: The Construction and Evaluation of a Syntactic Calculus

This paper discusses methods for presenting syntactic information in the form of a calculus, and for measuring its goodness of fit to a language.' Given a morpheme list for a language, the aim of a syntactic system is to tell us how to put together the sequences of morphemes which are used as sentences in the language. Such directions we shall call the FORMATION RULES. Additionally, works on syntax usually give a certain amount of information about the equivalences between some sequences and others, e.g. that John discovered the path = The path was discovered by John. We shall call such statements TRANSFORMATION RULES. Only a small part of the transformation rules is covered in syntax; some others are discussed in mathematical logic, e.g. Russell's theory of descriptions, procedures of generalization and abstraction. So far in mathematical logic, most attention has been directed to developing formation and transformation rules of artificial languages, and no complete treatment has yet been made of the transformation rules of a natural language. The division between formation and transformation rules can to some extent be altered by altering the units of the system. We shall concentrate on syntax as formation rules, and give a procedure for putting this material into a precise form so that it may be evaluated.2 Let us consider the syntax of a language L, say that of the English statement-type sentences, which may be constructed out of some finite word list. (For a con-